General Topology Meets Model Theory, on P and T M

General topology meets model theory, on p and t M. Malliaris ∗ and S. Shelah † ∗Department of Mathematics, University of Chicago, Chicago, IL 60637, USA, and †Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem 91904, Israel and Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

Submitted to Proceedings of the National Academy of Sciences of the United States of America

Cantor proved in 1874 that the continuum is uncountable, and The study of cardinal invariants of the continuum is a flourishing Hilbert’s first problem asks whether it is the smallest uncount- area, which lies at the intersection of set theory and general topology; able cardinal. A program arose to study cardinal invariants of the some properties reflect ideas from measure theory, algebra, or com- continuum, which measure the size of the continuum in various binatorics. The subject’s growth and development from the 1930s ways. By Godel¨ 1939 and Cohen 1963, Hilbert’s first problem is and 40s, in part a response to Hilbert’s problem, to the present can be independent of ZFC. Much work both before and since has been done on inequalities between these cardinal invariants, but some seen in the surveys of van Douwen 1984 [5], Vaughan 1990 [6], and basic questions have remained open despite Cohen’s introduc- Blass 2009 [4]. tion of forcing. The oldest and perhaps most famous of these is By Godel- 1939 [7] and Cohen 1963 [8], Hilbert’s first problem ℵ whether “p = t,” which was proved in a special case by Rothberger (whether 2 0 is the first uncountable cardinal) is independent of ZFC. 1948, building on Hausdorff 1934. In this paper we explain how In light of this, the study of cardinal invariants of the continuum be- our work on the structure of Keisler’s order, a large scale classifi- comes especially fertile. In most cases there are obvious ZFC results cation problem in model theory, led to the solution of this problem and usually independence proofs are deep and hard (using forcing, in ZFC as well as of an a priori unrelated open question in model of Cohen). ZFC answers to problems which have remained open for theory. some time are rare, and so, surprising. The problem of “p and t” appears throughout the literature. In Unstable model theory | Keisler’s order | cardinal invariants of c | p and t the seminal survey paper mentioned above, van Douwen catalogued and consolidated much of the work on these cardinals prior to 1984. He focused on six primary invariants: b, p, t (attributed to Rothberger Introduction 1939 [9] and 1948 [10]), d (attributed to Katetov 1960), a (attributed In this paper we present our solution to two long standing, and a priori to Hechler 1972 and Solomon 1977) and s (attributed to Booth 1974). unrelated, questions: the question from set theory/general topology As of Vaughan 1990 [6], only two inequalities about van Douwen’s of whether p = t, the oldest problem on cardinal invariants of the cardinals remained open: that of a and d and that of p and t, and “we continuum, and the question from model theory of whether SOP2 is believe [whether p < t is consistent with ZFC] to be the most inter- maximal in Keisler’s order. Before motivating and precisely stating esting” ([6] 1.1). Following Shelah’s [11] proof of the independence these questions, we note there were two big surprises in this work: of a < d, attention focused on whether “p = t” as both the oldest and first, the connection of the two questions, and second, that the ques- the only remaining open inequality about van Douwen’s diagram. tion of whether “p < t” can be decided in ZFC. We now define p and t. Let A ⊆∗ B mean {x : x ∈ A, x∈ / B} There have been few connections between general topology and ℵ0 is finite. Let D ⊆ [N] be a family of countable subsets of N. Say model theory, and these were exclusively in model theory: a key ex- that D has a pseudo-intersection if ∃A ⊆ N, A infinite such that ample is Morley’s use of ideas from general topology such as the (∀B ∈ D)(A ⊆∗ B). Say that D has the strong finite intersection Cantor-Bendixson derivative in proving his 1965 categoricity theo- property, or s.f.i.p., if every nonempty finite subfamily of D has infi- rem [1], the cornerstone of modern model theory. Here we have ap- nite intersection. Finally, say that D is a tower if it is linearly ordered plications in the other direction: by our proofs below, model theoretic by ⊇∗ and has no infinite pseudo-intersection. methods solve an old problem in general topology. It seems too early Definition 1. The cardinal p is the minimum size of a family F ⊆ to tell what further interactions will arise from these methods, but it [ ]ℵ0 such that F has the s.f.i.p. but no infinite pseudo-intersection. is a good sign that Claim 1 below will be generalized to P( )/ fin, N N The cardinal t is the minimum size of a tower. see Discussion 21. Full details of work presented here are available in our manuscript [2]. It is easy to see that p ≤ t, since a tower has the s.f.i.p. Hausdorff proved in 1934 [12] that ℵ1 ≤ p, and Rothberger proved in 1948 (in our terminology) that p = ℵ1 =⇒ p = t. This begs the question: Question 2. Does p = t? Cardinal invariants of the continuum After Rothberger [10], there has been much work on p and t, as Cantor’s 1874 proof of the uncountability of the continuum [3] fo- noted in the surveys [5], [6], [4], the introduction to our paper [2]; cuses attention on the region between ℵ0, the cardinality of the nat- see also the references [13], [14], [15], [16], [17] below. As noted, ℵ0 ural numbers, and the continuum 2 , the cardinality of the reals. A given the length of time the problem remained open, there was wide productive way to map this region is to consider properties of fam- confidence of an independence result. ilies of subsets of N which hold for any countable family and fail for some family of size continuum. For any such property of interest, call the minimum size κ of a family for which the property may Reserved for Publication Footnotes fail a cardinal invariant (or characteristic) of the continuum. The- orems about the relative sizes and interdependence of such cardinal invariants give fundamental structural information about families of subsets of N, and give precise ways to test the strength of countabil- ity hypotheses. As a simply stated example of such a cardinal, the bounding number b is the least size of a family F ⊆ ωω which is not bounded, meaning that there is no single g ∈ ωω which eventually dominates all f ∈ F , i.e. f ≤∗ g for all f ∈ F . As an example of results, the Cichon« diagram, [4] p. 424, gives implications between cardinal invariants relating primarily to measure and category. www.pnas.org/cgi/doi/10.1073/pnas.0709640104 PNAS Issue Date Volume Issue Number 1–7 Model theory Theorem B. (Shelah 1978 [19]) A complete theory T is either: From Morley’s theorem to Keisler’s order. Our solution to Question • <κ(T ) 2 arose in the context of our work on a model-theoretic classification stable, i.e. T is stable in all λ = λ , or • program known as Keisler’s order. As a frame for these investiga- unstable, i.e. T is unstable in all λ. tions, we briefly discuss model theory, stability, and ultraproducts. Under this analysis, the stable theories are “tame” or well- Model theory begins with the study of models, which are given by behaved, admitting a strong structure theory, while the unstable the- the data of an underlying set along with an interpretation for all func- ories are “wild.” tion, relation, and constant symbols in a fixed background language Discussion 5. Moreover, stability is local: T is unstable if and only L; (complete) theories, sets of L-sentences of first order logic true in if it contains an unstable formula, i.e. a formula with the order prop- some L-model; and elementary classes, classes of L-models which erty. Also, instability hides a structure/randomness phenomenon: share the same theory, written T h(M) = T h(N) or M ≡ N. any unstable theory either contains a configuration close to a bipar- It is well known that any two algebraically closed fields of the tite random (Rado) graph, or one close to a definable linear order. same characteristic and the same uncountable size are isomorphic. Los could not find any countable first order theory which had one Meanwhile, Keisler had proposed a very different way of clas- model, up to isomorphism, in some but not every uncountable cardi- sifying theories (and ultrafilters), from an asymptotic (ultrapower) nality, and so conjectured there was none. The cornerstone of modern point of view. It is of particular interest that Keisler’s order gives an model theory is the affirmative answer: outside definition of the stable theories, see Discussion 10 below. Theorem A. (Morley 1965 [1]) If T is a countable theory then either T is categorical [=has one model, up to isomorphism] in every An asymptotic classification.Keisler’s order, which we now ex- uncountable cardinal or T is categorical in no uncountable cardinal. plain, studies the relative complexity of theories according to the dif- Proofs of Morley’s theorem, due to Morley and later to Baldwin- ficulty of ensuring regular ultrapowers of some M |= T are saturated. Lachlan [18], show, in an abstract setting, that whenever a complete Definition 6. The filter D on I, |I| = λ is called regular if it contains theory in a countable language is categorical in some uncountable a regularizing family, i.e. X = {Xi : i < λ} ⊆ D such that any cardinal then its models admit analogues of many properties famil- infinite subset of X has empty intersection. iar from algebraically closed fields: for instance, the existence of It is consistent that all nonprincipal ultrafilters are regular. More- prime models, a well-defined notion of independence generalizing over, if D is regular and the background language L is countable then algebraic independence in algebraically closed fields, and existence M ≡ N =⇒ M λ/D is λ+-saturated iff N λ/D is, so the quantifi- of a maximal independent set whose size determines the model up to cation over all models in Definition 7 below is justified. isomorphism. Often, model theoretic arguments extract local struc- An ultraproduct of models hMi : i ∈ Ii with respect to the tural information from global constraints on a given elementary class ultrafilter D on I is the structure obtained by identifying elements (e.g. the property of having one model, up to isomorphism, in some Q of the product i Mi which are “D-almost everywhere” equivalent uncountable size). [20]. An ultrapower is an ultraproduct in which all the models Mi Model theory since Morley has developed in a number of ways; are the same. Ultrapowers are large (and, in some sense, generic) a central theme (and one crucial for our work) is stability, which we extensions of a given model M which remain in its elementary class. now describe. Let M be a model and A ⊆ M. The [1-]types over Their structure reflects and depends on the choice of ultrafilter D in A are the maximal consistent sets of formulas ϕ(x) in one free vari- subtle ways which are not yet well understood. However, precisely able with parameters from A. Informally, types describe elements this dependence suggests it is informative to compare two theories which may or may not exist in a given model M but which will al- by comparing ultrapowers of their respective models built using the ways exist in some extension of M, e.g. a transcendental element same ultrafilter. which will exist in field extensions of alg.A realization of a type Q Definition 7. (Keisler’s order, 1967 [21]) Let T1,T2 be countable p in a model M is an element a such that ϕ(a) is true in M for all theories. We say that T1 E T2 if: for all infinite λ, all D regular ϕ ∈ p. When λ is an infinite cardinal and the model M contains λ + on λ, all M1 |= T1, all M2 |= T2, if (M2) /D is λ -saturated then realizations of all types over all subsets of M of size < λ, we call M (M )λ/D is λ+-saturated. λ-saturated. For instance, an algebraically closed field of fixed char- 1 acteristic is λ-saturated if and only if it has transcendence degree at Problem 1. Determine the structure of Keisler’s order. least λ. Alternatively and informally, if a model M is considered as Keisler’s order is a problem about constructing ultrafilters on the sitting inside a large universal domain M, the 1-types over A ⊆ M one hand, and understanding theories on the other hand. We now give are the orbits under automorphisms of M fixing A pointwise, and some intuition for this problem and some of the main developments; M is λ-saturated if it contains representatives of all orbits under all for further details, see e.g. [22]. automorphisms of M fixing some set A ⊆ M, |A| < λ pointwise. The power of Keisler’s order comes from the fact that we may <ℵ We now define “stability” in a given cardinal λ, which led to a regard types as maps from [λ] 0 into D. Los’s theorem asso- fundamental structural dichotomy between theories which are stable ciates to each type p over A ⊆ M λ/D, |A| ≤ λ a monotonic [i.e. in many λ and those which are never stable, Theorem B below. u ⊆ v =⇒ f(v) ⊆ f(u)] map f :[p]<ℵ0 → D given by:

Definition 3. T is λ-stable if there are no more than λ types over any {ϕi0 (x; ai0 ), . . . , ϕin (x; ain )} 7→ M |= T , |M| = λ. ^ {t ∈ λ : M |= ∃x ϕ (x; a [t])} Example 4. If M is an algebraically closed field, the type of an el- i` i` ement over M is determined by the polynomials over M it does or `≤n does not satisfy; this gives |M| types [“stable”]. If M = (Q, <), the When D is regular, such a map can be chosen to be multiplicative, |M| types over M include all cuts; this gives 2 types [“unstable”]. If i.e. f(u ∪ v) = f(u) ∩ f(v), precisely when the type p is realized, M is the Rado graph, the types over M include all partitions of the see e.g. [22] §1.2. Moreover, when D is regular, we may choose the vertex set of M; this gives gives 2|M| types [“unstable”]. map so that its image is a regularizing family. The remarkable fact is that the stability or instability of mod- Thus, in certain cases, the problem of ensuring saturation in ultra- els resolves into a very informative classification of theories (as powers is directly reflected in a property of the underlying ultrafilter. in Morley’s theorem above, much of the information comes from Definition 8. (Keisler) D is λ+-good if every monotonic function the structural analysis which goes into the proof). In Theorem B, f :[λ]<ℵ0 → D has a multiplicative refinement g, i.e. u, v ∈ [λ]<ℵ0 κ(T ) ≤ |T |+ is an invariant of the theory. implies f(u) ⊇ g(u) ∈ D and g(u) ∩ g(v) = g(u ∪ v).

2 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author Good regular ultrafilters exist on all infinite λ by a theorem of The very interesting idea that Keisler’s order may further illuminate Keisler under GCH [23] and of Kunen in ZFC [24]. If D is regular randomness/structure tradeoffs within instability is supported by our and λ+-good then M λ/D is therefore λ+-saturated for any M in a Theorem 3 below. countable language. This is half of proving that Keisler’s order has Working together, Malliaris and Shelah have very recently made a maximum class. In fact, in the paper defining the order E, Keisler showed: significant progress on the problem of Keisler’s order in the 2012 papers [22], [30], [31], and [32]. These theorems are not prerequisites Theorem C. (Keisler 1967 [21]) for the current proofs, so we refer the interested reader to the introduction of [31] or [32] for more details. However, this work provided Tmin E ··· ??? ··· E Tmax strong motivation for addressing the long latent Question 16, which i.e. Keisler’s order has a minimum class and a maximum class. we now describe and solve in connection with “p = t”. Example 9. It follows from Theorem A above that any uncountably categorical theory, e.g. the theory of algebraically closed fields of The current approach: , , and SOP a fixed characteristic, is minimum in Keisler’s order. On the other p t 2 hand, one can show that any sufficiently rich theory, e.g. number the- In this section we motivate and announce the main results of our ory, will code any failures of goodness into a non-realized type and present work [2], with some comments on the proof. will therefore be maximal. Orders, trees, and the Keisler-maximal class. As described above, The only Keisler-equivalence classes characterized to date were the problem: given by: Problem 13. Give a model-theoretic characterization of the maxi- Theorem D. (Shelah 1978 [19].VI) mum class in Keisler’s order. is an old important problem in model theory, for which we know T1 / T2 / ··· ??? ··· E Tmax lower bounds and upper bounds: instability on one hand, and the where T1 ∪ T2 is precisely the stable theories, and: strict order property, or even SOP3, on the other (Discussion 11). The main theorems of our present work arose as part of our solution • T1, the minimum class, consists of the theories without the finite to Question 16 below, which moves the conjectured boundary of this cover property (f.c.p.). class onto what appears to be a major dividing line for which there • T2, the next largest class, consists of stable theories with f.c.p. are strong general indications of a theory. • Tmax contains all linear orders, e.g. T h(Q, <), though its Before explaining this comment, let us define: model-theoretic identity is not known. [Indeed, the weaker order Definition 14. The formula ϕ has SOP2 with respect to a theory property SOP3 suffices for maximality, by Shelah 1996 [25].] T if in some sufficiently saturated M |= T there are parameters ω> `(y) {aη : η ∈ 2} ⊆ M such that: Discussion 10. In Theorem D the class of stable theories, charac- ω> terized in terms of counting types in Theorem B, is independently for each 1 ≤ n < ω and and η1, . . . ηn ∈ 2, characterized in terms of saturation of ultrapowers. {ϕ(x; aηi ) : 1 ≤ i ≤ n}

Discussion 11. From Theorems C and D we see that although is consistent iff η1, . . . ηn lie along a single branch. We say the theory Keisler’s order has a maximum class, we have only weak upper and T has SOP2 if some ϕ does, with respect to T . lower bounds on properties for membership in this class. Example 15. In (Q, <) the formula ϕ(x; y, z) = y < x < z has In the intervening years, progress was slow, due in part to the SOP2; SOP2 also arises in the generic Kn-free graph for any perceived complexity of ultrafilters. Beginning in 2009, Malliaris n ≥ 3, a theory well into the “independent” region of instability. advanced the problem of how ultrafilters and theories interact in a se- Recall from Theorem D that any theory which includes a defin- ries of papers [26], [27], [28], [29], motivated by Discussion 10. That able linear order, e.g. (Q, <), or even just SOP3 (which we shall is, given that Keisler’s order independently detects the jump in com- leave as a black box; see [25] for details) , is maximal. Whereas plexity between stable and unstable theories, the goal was to leverage SOP3 retains many features of linear order, SOP2 describes a kind Keisler’s order to describe gradations in complexity among the un- of maximally inconsistent tree so did not appear amenable to the stable theories from a more uniform point of view. See Malliaris [29] same kinds of arguments. We have the implications: for a further discussion. For the present work, we quote: not stable ← SOP2 ← SOP3 ← strict order property Theorem E. (Malliaris 2009 [26]) For regular ultrapowers in a countable language, local saturation implies saturation. That is, Following Shelah 1996 [25] the key question, which we solve below, Keisler’s order reduces to the study of ϕ-types. became: Question 16. Is any theory with SOP maximal in Keisler’s order? Theorem F. (Malliaris 2010 [28]) There is a property of filters, 2 called flexibility, which is detected by any theory which is non-low Why is the move from SOP3 to SOP2 so significant? First, there (some formula k-divides with respect to arbitrarily large k). is the nontrivial matter of developing a framework to compare orders Theorem G. (Malliaris 2010 [28]-[29]) There is a minimum Keisler and trees. Second, recent evidence (in work in preparation of the au- class among the theories with TP2, whose saturation is character- thors) suggests that SOP2 may characterize maximality, Conjecture ized by existence in D of internal maps between small sets, and by 29 below. the ultrafilter D being “good for equality”. In order to set the stage for the framework which led to the solu- Discussion 12. Keisler’s order imposes a hierarchy on the struc- tion of both Questions 16 and 2, we present our strategy for solving ture/randomness phenomenon from Discussion 5: the “structured” Question 16. The first key idea was to describe the realization of (in some sense, rich) theories, those with linear (strict) order, are SOP2-types in terms of a property of an ultrafilter D, which could Keisler-maximum whereas the “purest” random theory, that of the then be compared against goodness for D, as by Theorem D above D λ Rado graph, is Keisler-minimum among the unstable theories. See is good iff (N, <) /D has no (κ1, κ2)-cuts for κ1 + κ2 ≤ λ. [29] or [22] §4. TP 2/SOP 2 is a randomness/structure phe- Definition 17. A subset Y of a partially ordered set (X, <) is cofinal nomenon in non-simple theories with analogies to the indepen- in X if (∀x ∈ X)(∃y ∈ Y )(x ≤ y). We use the word coinitial when dence/strict order phenomenon in non-stable theories. Theorem G the order is reversed. By the cofinality of a partially ordered set we shows that TP2, the “random side” of this more complex example, mean the smallest size of a cofinal subset. Call an infinite cardinal κ also admits a Keisler-minimum class, of a particularly simple form. regular if, considered as an ordinal, its cofinality is κ.

Footline Author PNAS Issue Date Volume Issue Number 3 Note: in this paper, we will have two different uses of the word Discussion 21. In work in progress, we have an analogue of Claim “regular”: as applied to ultrafilters, Definition 6 above, and as applied 1 for P(N)/ fin. to cardinals, Definition 17. Before describing our approach in full generality, we recall that Definition 18. Let κ, θ be infinite regular cardinals and (A, <) a the hypothesis p < t also connects to questions about cuts. total linear order. We say there is a (κ, θ)-cut or gap in A when we can write A = A1 ∪ A2 such that (a) A1 ∩ A2 = ∅, (b) Peculiar cuts in p and t. Here we describe a further useful ingredi- (∀a ∈ A1)(∀b ∈ A2)(a < b), (c) there is a cofinal κ-indexed se- ent which connects p < t to the appearance of a certain kind of cut quence in (A1, <) and (d) there is a coinitial θ-indexed sequence in (gap), Definition 18. The ∗ in ≤∗ means “for all but finitely many”. (A2, <). So in our terminology, cuts are not filled. Definition 22. (Peculiar cuts) For κ1, κ2 infinite regular cardinals, a Definition 19. The regular ultrafilter D on λ has κ if: when- ω -treetops (κ1, κ2)-peculiar cut in ω is a pair (hfα : α < κ1i, hgβ : β < κ2i) ever M = (T , ) is a tree and N = M λ/D its ultrapower, any ω 0 E of sequences of functions in ω such that: (a) α < α < κ1 -increasing sequence in N of cofinality < κ has an upper bound. 0 ∗ ∗ ∗ E and β < β < κ2 implies fα ≤ fα0 ≤ gβ0 ≤ gβ ; (b) ∗ ∗ We now have the following measure of the complexity of D: (∀α < κ1)(fα ≤ h) implies (∃β < κ2)(gβ ≤ h); and (c) ∗ ∗ (∀β < κ1)(h ≤ gβ ) implies (∃α < κ1)(h ≤ fα). C(D) = {(κ1, κ2): κ1, κ2 regular, κ1 + κ2 ≤ |I|, Theorem H. (Shelah [33]) Assume p < t. Then for some regular car- I ω and (N, <) /D has a (κ1, κ2)-cut} dinal κ there exists a (κ, p)-peculiar cut in ω, where ℵ1 ≤ κ < p. recalling that C(D) = ∅ iff D is good (“maximally complex,” that Looking ahead, in the context of forcing, we will connect Theo- is, able to saturate ultrapowers of linear order, which belong to the rem H to a general version of the “cut spectrum” C(D), as follows: Keisler-maximum class). Example 23. If G ⊆ [ω]ℵ0 is a generic ultrafilter then writing Q We have therefore translated Question 16 to: X = g0(n), we have that hfα/G : α < κ1i, hgβ /G : β < κ2i + G Question 20. Suppose D is a regular ultrafilter on λ with λ - give a (κ1, κ2)-cut in X. treetops. Is C(D) = ∅? To illustrate some simple properties of the cofinality spectrum + A common context for both problems. The right framework, dis- C(D) under λ -treetops, and to motivate the later abstraction of “co- covered in [2], is more general than regular ultrapowers; however, in finality spectrum problems,” we now sketch the proof of a uniqueness defining it we draw on intuition from the proof of Claim 1 sketched result in this context. above. The abstraction captures the fact that when studying regular Claim 1. ([2] Claim 3.3, in the special case of ultrapowers) Let D ultrapowers of (N, <), since ultrapowers commute with reducts, we be a regular ultrafilter on λ, κ = cf(κ) ≤ λ, and suppose D has may assume that the nonstandard copy of in the ultrapower N con- + N λ -treetops. If (κ, θ0) ∈ C(D) and (κ, θ1) ∈ C(D), then θ0 = θ1. tinues to behave in a “pseudofinite” way (e.g. all nonempty definable λ N N Proof. Let M = (N, <) and let N = M /D. Suppose that in subsets of N have what N believes to be an < -least element, and 0 0 N N, (haα : α < κi, hb : < θ0i) represents a (κ, θ0)-cut while all such bounded subsets have a < -greatest element). Moreover, 1 1 (haα : α < κi, hb : < θ1i) represents a (κ, θ1)-cut. we may expand the language in order to have available uniform defi- As ultrapowers commute with reducts, we may suppose, in an ex- nitions for relevant trees. Recall Definition 18 above. panded language [e.g. expanding M to (H(ℵ1), )], that M includes Definition 24. (Informal; for full details, see [2] Definition 2.1 pps. a definable tree T of finite sequences of pairs of natural numbers, 7–8) Informally, we say that the triple M, M1, ∆ has “enough set strictly increasing in each coordinate, with E denoting a definable theory for trees” when: M M1; ∆ is a nonempty set of formulas partial order on T corresponding to initial segment. Identifying N defining discrete linear orders in M1, such that every nonempty M1- N N with its induced expansion to the larger language, let (T , E ) de- definable subset of any such order has a first and last element; and note the D-ultrapower of this tree. For ease of reading, write c(n, i) for each instance of a formula ϕ in ∆ (i.e. for each order) there is a for the ith coordinate of c(n). definable tree of sequences, i.e. funcitons from the order to itself. N By induction on α < κ, we choose cα ∈ T and nα = We ask that for each such tree, length, projection, concatenation, i max dom(cα) such that cα(nα, i) = aα for i = 0, 1. The base and the maximal element of the domain of a given function, and the par- successor case are easily done. In the limit case, we will have defined tial ordering E are definable; and we ask that ∆ be closed under a sequence hcβ : β < αi of elements lying along a single branch in Cartesian products. N N T . The assumption of treetops gives an upper bound c∗ ∈ T Our true context is therefore the following: for this sequence, i.e. cβ E c∗ for β < α. Then, in our expanded 0 1 Definition 25. ([2] 2.5–2.11) We say that s is a CSP (cofinality spec- language, the set {n ≤ n∗ : c∗(n, 0) < aα ∧ c∗(n, 1) < aα} is a trum problem) when: nonempty, definable, bounded subset of N , and so it has a maximal + + N s = (M,M1,M ,M1 , ∆), where ∆ is a set of formulas of L(M), a 0 1 element m. To complete this step, set cα := c∗ m (aα, aα). + + M M1, and this pair can be expanded to M M1 such that Having completed the definition of the sequence hcα : α < κi, + + + (M ,M1 , ∆) has enough set theory for trees. When M = M or we may again by treetops find an upper bound for this sequence, + N M1 = M , we may omit it. c∞ ∈ T . Let n∞ = max dom(c∞). For i ∈ {0, 1} and < θi, 1 i i In this context we define: let n = max{n ≤ n∞ : c∞(n, t) < b}. This is again a nonempty, definable, bounded subset of the nonstandard natural numbers, so s s 0 1 1. Or(s) = Or(∆ ,M1 ) is the set of orders defined in M1 by in- each n , n are well defined. But now we have effectively “sewn i stances of formulas from ∆ together” the two original cuts. The sequences hb : < θi and 2. Cct(s) = {(κ , κ ): there is a ∈ Or(s,M ) such that the linear i 1 2 1 hc(n, i): i < θi are mutually cofinal for i = 0, 1. Suppose for order ≤a on Xa has a (κ1, κ2)-cut } N 0 a contradiction that, in N , the sequences (hnα : α < κi, hn : Note: By definition of cut in Defn. 18 above, κ1, κ2 are regular. 1 < θ0i) and (hnα : α < κi, hn : < θ1i) do not describe the 3. Tr(s) = {Ta : a ∈ Or(s)} is the set of definable trees associated 0 same cut. Without loss of generality, hn : < θ0i is not cofinal to orders in Or(s) 1 ttp in hn : < θ1i. So there is some ∗ < θ1 such that [recalling we 4. C (s) = {κ : κ ≥ ℵ0, a ∈ Or(s) and there is in the tree Ta 1 have strict monotonicity in each coordinate] c∞(n∗ , 0) lies in what a strictly increasing sequence of cofinality cf(κ) with no upper 0 0 we assumed to be a cut described by (haα : α < κi, hb : < θ0i), bound } ttp a contradiction. Since we assumed θ0, θ1 are regular, θ0 = θ1, as 5. Let ts be min C (s) and let ps be min{κ :(κ1, κ2) ∈ ct desired. • C (s) and κ = κ1 + κ2}.

4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author ct ttp Note that by definition of C (s) and C (s), both ts and ps are Finally, by the Second Reduction described above, Theorem 2 regular. gives: Theorem 6. ([2] Theorem 5.17) We focus on C(s, ts) where this means: p = t ct 6. C(s, ts) = {(κ1, κ2): κ1 + κ2 < ts, (κ1, κ2) ∈ C (s)}. These results give insight into other structures, as well. For instance, it is a corollary of these methods that in any model of Peano First Reduction. If for every CSP s, C(s, ts) = ∅ then SOP2 is maximal in Keisler’s order. Why? This simply translates Question 20 to arithmetic, if for all κ < λ the underlying order has no (κ, κ)-cuts this more general context: the issue is whether knowing there are al- (gaps) then the model itself is λ-saturated. ways paths through trees is sufficient to guarantee that there are no cuts of small cofinality. Key steps in the proof. Here we state the intermediate claims in the However, our new context also encompasses other problems. proof that C(s, ts) = ∅, Theorem 2 above. [For the connection to p, t and SOP2, see the sections “First Reduction” and “Second Reduc- Second Reduction. To deal with p and t in this context we look for tion” above.] a cofinality spectrum problem, and for a cut, recalling Fact H above. Claim 4. (Uniqueness, [2] Claim 3.3) Let s be a cofinality spectrum Suppose we are given M = (H(ℵ1), ), a forcing notion Q = problem. Then for each regular κ ≤ ps, κ < ts: ([ω]ℵ0 , ⊇∗), V a transitive model of ZFC, and G a fixed generic sub- 1. (κ, λ) ∈ Cct(s) for precisely one λ. Q P(ω)V set of , forced to be an ultrafilter on the Boolean algebra . 2. (κ, λ) ∈ Cct(s) iff (λ, κ) ∈ Cct(s). Consider N = M ω/G in the forcing extension V[G]. Recall that first, the diagonal embedding is elementary, so M N; and This proof amounts to a generalization of Claim 1 above to the second, Q is t-complete so forcing with Q adds no new bounded general context of cofinality spectrum problems. V V V[G] V[G] subsets of t, and p < t iff p < t . Claim 5. (Anti-symmetry, [2] Claim 3.45) Let s be a cofinality spec- We focus attention on the cofinality spectrum problem s∗ = trum problem. Then for all regular κ such that κ ≤ ps, κ < ts, we (M,N, ∆psf ) where ∆psf is the set of definable linear orders in have that (κ, κ) ∈/ Cct(s). N which are covered by G-ultraproducts of finite linear orders. We shall assume p < t and prove several claims. The proof involves constructing a path through a tree of pairs of natural numbers, in the spirit of Claims 1 and 4. Claim 2. ([2] 5.8, assuming p < t) t ≤ ts∗ . N Corollary 27. (c.f. [2] Cor. 3.9) Let s be a cofinality spectrum prob- Proof Sketch. Without loss of generality, work in the tree T = lem. In light of the above we may, without loss of generality, study (ω>ω, )N hf˜ /G : α < θi N θ < t E . Let α be a proposed path in , . C(s, ts) by looking at ˜ N N Let B “hfα/G : α < θi is E -increasing in T ”, and with- ˜ {(κ , κ ):(κ , κ ) ∈ C(s, t ), κ < κ } out loss of generality B “fα = fα” as there are no new se- 1 2 1 2 s 1 2 n quences of length < t. For each n ∈ B, look at the cone Yα M S n Corollary 27 summarizes the two previous claims. above fα(n) in in the index model T . Let Yα = n∈B Yα . Then α < β =⇒ Yβ ⊆∗ Yα, as otherwise we contradict the choice of Claim 6. (On lcf(ℵ0), [2] Claim 3.58) Let s be a cofinality spec- B. Thus {Yα : α < θ} is a tower of length θ < t, and by definition trum problem and suppose lcf(ℵ1, s) ≥ ℵ2. Then (ℵ0, ℵ1) ∈/ of t as the tower number, there is an infinite Y with Y ⊆∗ Yα for all C(s, ts) =⇒ (ℵ0, λ) ∈/ C(s, ts) for all regular λ ≤ ps, λ < ts. α < θ. From any such Y it is easy to build an upper bound for our The proof of Claim 6 is an order of magnitude more compli- given path. • cated, and requires using (or developing) more general trees; so as to always have these available, we prove from the definition that cofi- Claim 3. ([2] 5.15, assuming p < t) ps∗ ≤ p. The proof shows, using Fact H and the definition of G, that if nality spectrum problems always have available a certain amount of ω Q Peano arithmetic. Claim 6 may be read as saying that the only case p < t then for some nonconstant g0 ∈ ω, G g0(n) contains a (κ, p)-cut where κ is regular, ℵ ≤ κ < p. of concern for ℵ0 is a “close” asymmetric cut, (ℵ0, ℵ1). Indeed, in 1 the general result, Lemma 1 below, the a priori most difficult case to Conclusion 26. ([2] 5.16) If p < t then C(s∗, ts ) 6= ∅. + + ∗ rule out was a (κ, λ)-cut when κ = λ and λ = ts. s C(s, t ) = ∅ p = t We have shown: if for every CSP , s , then . Lemma 1. (Main lemma, [2] Lemma 3.65) Let s be a cofinality spec- This completes the second reduction. trum problem. Suppose that κ, λ are regular and κ < λ = ps < ts. Then (κ, λ) ∈/ C(s, ts). Our main theorems. In Malliaris and Shelah 2012 [2] we prove the following. The first, main structure theorem is Theorem 2, proved by This lemma, which supercedes and substantially generalizes model-theoretic means; some key lemmas are described in the next Claim 6, is the core of the argument. Its proof requires formaliz- section. ing a robust picture of how sets of large size may be carried along one side of a cut to overspill into the other when the cofinality on Theorem 2. ([2] Theorem 3.66) For any CSP s, C(s, ts) = ∅. either side is small relative to the size of “treetops”. We also show By the First Reduction described above, Theorem 2 gives: that cofinality spectrum problems have available an internal notion of cardinality. So for suitably chosen representations of cuts such over- Theorem 3. ([2] Theorem 4.48) SOP2 is maximal in Keisler’s order. spill is enough for a contradiction, which rules out the existence of By some further arguments in the case of ultrafilters, our methods cuts of the given cofinality. [2] 3.61 is an informal description of the and Theorem 2 also give a new characterization of Keisler’s notion proof. of goodness: With these results in hand we prove the paper’s fundamental re- Theorem 4. ([2] Theorem 4.49) For a regular ultrafilter D on λ, (a) sult: + κ ≤ λ =⇒ (κ, κ) ∈/ C(D) iff (b) C(D) = ∅ iff (c) D is λ -good. Proof. (of Theorem 2) If ts ≤ ps, then by definition of ps, C(s, ts) = ∅. So we may assume ps < ts. Let κ, λ be such that By Theorem 3 and Theorem G above, we obtain: κ + λ = ps and (κ, λ) ∈ C(s, ts). By Claims 4, 27 and 5, κ 6= λ and Theorem 5. ([2] Theorem 4.51)The minimum TP2 class is the min- we may assume κ < λ = ps. Then the hypotheses of Lemma 1 are imum non-simple class in Keisler’s order. satisfied, so (κ, λ) ∈/ C(s, ts). This completes the proof. •

Footline Author PNAS Issue Date Volume Issue Number 5 Discussion 28. As regards Keisler’s order, the work presented here ACKNOWLEDGMENTS. Malliaris was partially supported by NSF grant DMS- is related to the non-structure side, giving sufﬁcient conditions for 1001666, by Shelah’s NSF grant DMS-1101597, and by a Godel¨ research fel- maximality. lowship. Shelah was partially supported by the Israel Science Foundation grants 710/07 and 1053/11. This is paper E74 in Shelah’s list of publications.

Conjecture 29. SOP2 characterizes maximality in Keisler’s order.

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